Showing papers on "Stackelberg competition published in 1978"

On generalized Stackelberg strategies

Abstract: The concept of Stackelberg strategy for a nonzero-sum two-person game is extended to allow for a nonunique “rational” response of the follower. This leads to the notion of a generalized Stackelberg strategy for the leader, which guarantees him a cost value that cannot be exceeded, no matter what the “rational” response of the follower. Then, a generalized Stackelberg strategy pair is defined. A simple example is given. The idea of a generalized Stackelberg strategy and strategy pair is then applied to the situation of one leader and many “rational” followers.

Leader-follower strategies for multilevel systems

Abstract: Sequential strategies for dynamic systems with multiple decision makers and multiple performance indices are surveyed and reviewed. These strategies are generalizations of Stackelberg or leader-follower strategies for two-person games. The review includes structures with one coordinator and several second-level decision makers, and linear hierarchical structures with only one decision maker at each level Several information structures are considered.

Dominant Firm Pricing Policy in a Market for an Exhaustible Resource

Abstract: The paper describes a von Stackelberg model of pricing behavior by a dominant firm in a market for an exhaustible resource. The results obtained differ dramatically from those that characterize a pure monopoly. If the marginal production cost in the competitive fringe is constant, the optimal dominant firm price strategy is independent of its own costs and is determined by the characteristics of the fringe. In contrast, if the production cost of the fringe is constant only up to a capacity constraint, the cartel may maximize profits by acting as a classical limit-pricing firm.

Closed-loop Stackelberg strategies in linear-quadratic problems

Abstract: Linear closed-loop Stackelberg strategies in sequential decision-making problems for linear systems and trace criteria which may be interpreted as the expected value of quadratic criteria when the initial state of the system is uniformly distributed over the unit sphere are considered. Necessary conditions for the solution are developed and an algorithm for numerical solution of two-level Stackelberg problems is presented. Generalization to multilevel Stackelberg problems is described and specific results for the three-level sequential decision-making problem are presented.

The Equilibrium of Duopoly in a Market of Homogeneous Goods

Abstract: In this paper, we consider duopoly equilibria in which firms act rationally on the basis of mutually consistent assumptions about their rivals' behaviour. It turns out that there exist various kinds of equilibria corresponding to various pairs of behaviour patterns. Depending on the relation between the firms' cost functions and the market demand functions there are different outcomes: monopoly, limit-pricing, stable asymmetric duopoly with one firm a price-maker and the other (less efficient) firm a price-taker, and unstable symmetric duopoly which might have price competition (as in Bertrand, 1883) or quasi-agreement (Fellner, 1960). There remain some weak points in the structure of traditional oligopoly theory as developed by Cournot (1838), Bertrand (1883), Edgeworth (1925), Stackelberg (1934) and others. Cournot hypothesizes that an oligopolist assumes rivals' quantity to be fixed, but this assumption typically turns out to be incorrect as the rivals' reactions occur, and Shapley and Shubik (1969) argue that price is the more natural strategic variable. Bertrand and Edgeworth, analysing duopolists' behaviour with price-setting strategies, find that there is price competition only because of a special assumption about cost conditions. Stackelberg considered asymmetric duopoly, with a leader and a follower, and concluded as follows. (a) There cannot be a "leader-follower" outcome in an oligopolistic market for a homogeneous good with price-setting strategies (see also Fellner, 1960, Chapter 2). The outcome must be price competition as in Bertrand and Edgeworth. (b) Quantity-strategy oligopoly almost always has a "leader-leader" outcome. In this paper we shall investigate rational behaviour of a duopolist who takes account of price as well as quantity in his strategy, and we shall examine various patterns of duopoly. In Section I we shall analyse rational behaviour of a duopolist who conjectures his rival's reaction with respect to price as well as quantity. In Section II, comparing his profit in various cases, we shall describe optimal behaviour. In Section III we shall show various kinds of duopoly equilibria that correspond to various cost curves of both duopolists. Rigorous mathematical arguments are set out in the Appendix.

Multilevel Stackelberg strategies in linear-quadratic systems

Abstract: Open-loop multilevel Stackelberg strategies in deterministic, sequential decision-making problems for continuous linear systems and quadratic criteria are developed. Characterization of the Stackelberg controls via the solution of a higher-order square-matrix-Riccati differential equation is established; also, the basic structural properties of the coefficient matrices of this differential equation are established, and the basic structural properties of its solution are inferred.

Closed-loop stackelberg strategies for singularly perturbed linear quadratic problems

Abstract: Linear closed-loop Stackelberg strategies for systems with slow and fast modes are considered. The asymptotic behavior of the solution is studied and used to derive near optimal strategies which does not require the knowledge of the small singular perturbation parameter.

A new approach for derivation of closed-loop stackelberg strategies

Abstract: This paper develops a new approach to obtain the closed-loop Stackelberg (CLS) solution of an important class of two-person nonzero-sum dynamic games characterized by linear state dynamics and quadratic cost functionals. The new technique makes use of an important property of nonunique representations of a closed-loop strategy, and it relates the CLS solution to a particular representation of the optimal solution of a team problem. This specific problem treated in this paper is a 3-stage Scalar LQ dynamic game, and within this context it is shown that the CLS strategies for the leader are linear and of the one-step memory type, while those of the follower can be realized in linear feedback form. Extensions of this solution to the general LQ dynamic two-person game are also discussed in the paper.

A sampled data equilibrium Stackelberg coordination scheme for the multicontroller problem

Abstract: In this paper we consider a multicontroller problem in which each control agent has a different objective function. The actions of the controllers are coordinated by the influence of one controller who plays according to a Stackelberg strategy relative to the remaining players. We investigate this formulation when the information flow is restricted by constraining it to be in the form of sampled data acquisition. The computational advantage of the sampled data formulation is quite significant. Of equal importance is the relationship among the sampled data, closed-loop, and open-loop problems. The existence of and solution for the closed-loop Stackelberg solution for the continuous-time game are, at present, unresolved problems. The primary motivation for considering the sampled data formulation is to obtain a solution which maintains the computational simplicity of the tractable open-loop solution while gaining the responsiveness of a state feedback solution, avoiding the complications of the purely closed-loop formulation. The linear quadratic problem is considered in detail, and an efficient solution algorithm is derived which takes advantage of certain characteristics of the sampled data solution.

Nash and Stackelberg solutions to general linear-quadratic two player difference games. I. Open-loop and feedback strategies

Abstract: Concepts needed in the definition of difference game problems are first studied in detail. Solutions for a general class of deterministic linear quadratic two-player nonzero-sum difference games are then developed. Nash and Stackelberg solutions for open-loop and zero-memory information structure are considered. An augmentation techniques and a dynamic programming approach are applied to obtain the solutions. A recursive algorithm is developed for the Nash open-loop solution. Computational difficulties, which are caused by the augmented representa­ tions when the number of time stages is great, are thus avoided.

The Dynamics of Some Discrete Time Duopoly Price Games Involving Cournot and Related Best-Response Strategies

Abstract: In 1838 Cournot stated the duopoly problem and proposed his solution based on output variation. Then Bertrand, in 1883, objected to Cournot's method, suggesting that in practice each duopolist will resort to price variation, assuming that the rival will keep his price constant. Thus competitive underbidding will eventually lead to a zero profit. Edgeworth (1925) showed that this theory leads to price oscillations between the monopoly price and a price just above zero. Nichol (1934) pointed out that Cournot was describing a double or multiple auction sale, although he did not explicitly outline his assumptions. Levitan and Shubik (1972) argued that the specifics of the market determine the strategic variable. Wicksell and Bowley were originally responsible for leading duopoly theory back towards an emphasis on output policy. Recently, output variation models have often been chosen for study in preference to price variation models, mainly because of their historical significance and the reduced computational and descriptive problems associated with them; see for example, Mayberry, Nash and Shubik (1953) and Osborne (1971). However, although Osborne originally considered output variation, he believed that price is the usual action variable and subsequently (1974) considered the corresponding, though not strategically equivalent, price variation model. In his duopoly price model, Osborne (1974) considered a noncooperative game with perfect and complete information and unrestricted communication. Firms know each others' past and current prices. Osborne proceeded to discuss the equilibrium situations arising from various combinations of strategies, namely Cournot, Stackelberg, market-share, Nichol, Bishop, equal-price and leadership price-fixing strategies. However, Osborne did not make it clear whether he was considering a continuous or discrete time model. Footnote 1, p. 161, of his price variation model (1974) suggests the latter but his mathematical analysis is valid only in the former case. Limitations on labour and machinery changes affect the frequency with which output can be varied. Although prices can sometimes be altered more readily than output, they too should realistically be regarded as subject to change at discrete time intervals rather than on a continuous basis. In this paper we consider in detail the Cournot price-fixing strategy, which we will find it convenient to denote SO, together with the set of strategies S_ n > 1, where Sn is defined as the best-response strategy to S.-,. Thus, for example, S, denotes the Stackelberg strategy. We should make it clear that we are not considering the strategies S_, n 2, merely because they have intrinsic appeal to us. Rather, as we shall see later, the

Nash and Stackelberg solutions to general linear-quadratic two-player difference games. II. Open-closed strategies.

Abstract: This paper deals with difference games in which the information structures of the players are not equal. Nash and Stackelberg solutions to deterministic linear-quadratic two-player difference games are presented. The open-closed strategies are considered where one of the players can only use the initial state whereas the other has access to perfect memory information of the state. Stackelberg open-closed strategies are treated only in the case where the leader has access to open-loop information. A combined dynamic programming and augmentation techni­ ques approach is developed. A computationall y straightforwar d recursive algorithm is derived for the Nash open-closed solution.